A015950 Numbers k such that k | 4^k + 1.

1, 5, 25, 125, 205, 625, 1025, 2525, 3125, 5125, 8405, 12625, 15625, 25625, 42025, 63125, 78125, 103525, 128125, 168305, 202525, 210125, 255025, 315625, 344605, 390625, 517625, 640625, 841525, 875125, 1012625, 1050625, 1275125

Offset: 1

Author

Robert G. Wilson v

Keywords

• nonn

Comments

From Robert Israel, Sep 14 2017: (Start)

All terms except 1 are congruent to 5 mod 20.

If k is a term and prime p | k, then k*p is a term.

All prime factors of terms == 1 (mod 4).

If p is a prime == 1 (mod 4) and the order of 4 (mod p) is 2*m where m is in the sequence, then m*p is in the sequence. (End)

Examples

			4^5 + 1 = 1025 and 1025 is divisible by 5, so 5 is a term.
		

Links

• Max Alekseyev, Table of n, a(n) for n = 1..3514 (first 325 terms from Robert Israel)

Crossrefs

Cf. A015945, A211349.

Column k=4 of A333429.

Programs

Magma

[n: n in [1..10^6] | Modexp(4, n, n)+1 eq n]; // Jinyuan Wang, Dec 29 2018

Maple

select(n -> 4 &^ n + 1 mod n = 0, [1, seq(i,i=5..10^7,20)]); # Robert Israel, Sep 14 2017

Mathematica

Select[Prepend[20 Range[0, 10^5] + 5, 1], Mod[4^# + 1, #] == 0 &] (* Michael De Vlieger, Dec 31 2018 *)

PARI

is_A015950(n) = Mod(4,n)^n == -1; \\ Michel Marcus, Sep 15 2017

Python

A015950_list = [n for n in range(1,10**6) if pow(4,n,n) == n-1] # Chai Wah Wu, Mar 25 2021